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Abstract

We introduce an efficient method for fully characterizing multimode linear-optical networks. Our approach requires only a standard laser source and intensity measurements to directly and uniquely determine all moduli and non-trivial phases of the matrix describing a network. We experimentally demonstrate the characterization of a 6×6 fiber-optic network and independently verify the results via nonclassical two-photon interference.

Figures (4)

Fig. 1 Scheme for characterizing a linear-optical network M. (a) Using a 50:50 beam splitter (BS) and phase shifter (ϕ) a dual-mode coherent state, |α〉, is prepared and sent through M, where |α2〉= |eiϕα1〉. By sequentially inputting |α2〉 into modes 2,3,...,N, and varying the phase over at least 2π, all phases of matrix M can be directly determined. (b) Experimental realization. The device-under-test is a 3×3-mode fused-fiber beam splitter (FBS), which constitutes a 6×6 optical network including polarization. Orthogonal polarization modes are resolved using fibre polarization beam splitters (FPBS) at its outputs. Interferometric probe states between pair-wise input combinations are prepared with two polarization beam displacers (BD), and half-wave plates (HWP). The outputs are monitored with fast photo-diodes (PD) connected to an oscilloscope (OSC) while the phase ϕ is scanned.

Fig. 2 Experimental characterization of a linear-optical device. (a) Moduli rjk of the experimentally measured M. The x and y axes correspond to the input and output modes, j and k respectively. (b) Representative experimental data for obtaining θjk when injecting the variable-phase dual-mode coherent state into input modes 1 and 3. The amplitudes (voltage at output photo-diodes) of the six output modes (1–6 from top-to-bottom) oscillate as the phase ϕ is swept in time. Red and blue lines are measured data and theoretical fits to Acos(ϕ − θjk) respectively. (c) Phases, θjk, of the measured matrix M. The entire characterization method was performed 10 times to obtain experimental uncertainties; error bars are not visible on the scales shown.

Fig. 3 Independent verification of the measured matrix M. (a) Experimental schematic. A pair of 820 nm photons is generated via type-I spontaneous parametric downconversion (SPDC) in a nonlinear β-barium-borate (BBO) crystal pumped with a mode-locked pulsed laser at 410 nm. After being spectrally filtered (FWHM 2 nm) individual downconverted photons are steered into the optical modes of the linear-optical network by a series of beam displacers (BD) and half-wave plates (HWP). The temporal overlap, Δt, between the input photons is controlled via a micro-translation stage at one of the inputs. Output photons are detected using avalanche photo-diodes (APD) whose coincident signals—monitored using a commercially available counting logic—are used to post-select two single photon events. (b) Measured nonclassical visibilities vs. predicted visibilities for photons input into modes {1, 3} and {1, 6}. Red bars show the directly measured nonclassical visibilities; blue bars show the predictions from the measured matrix M; errors are given at the top of each data point. Numbers on the x-axis show the corresponding output modes.

Fig. 4 (a) Scheme for characterizing path-independent loss in the linear optical network. Virtual beam splitters are placed at each input mode of the optical network represented by a lossless matrix B. The resulting matrix V is a 2N×2N matrix and accounts for path-independent loss. (b) The matrix VV†. The diagonal hatched squares are equal to unity by construction and the off-diagonal elements are coloured according to their value given by the color bar.